Dynamical wetting transitions : liquid film deposition and air entrainment

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liquid film deposition and

air entrainment

Tak Shing Chan

Dynamical wetting

transitions:

liquid film deposition and

air entrainment

Dynamical wetting transitions:

liquid f

ilm de

position and

air entr

ainment

liqu id film de po sitio n and air ent rainmen t Ta k Sh ing C han

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Dynamical wetting transitions: liquid film deposition and air

entrainment

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Samenstelling promotiecommissie:

Prof. dr. Leen van Wijngaarden (voorzitter) Universiteit Twente Prof. dr. Detlef Lohse (promotor) Universiteit Twente Dr. Jacco H. Snoeijer (assistant promotor) Universiteit Twente

Prof. dr. Bruno Andreotti ESPCI, Paris

Prof. dr. Jens Eggers University of Bristol

Mr. Michel Riepen ASML

Prof. dr. Paul Kelly Universiteit Twente

Prof. dr. Serge Lemay Universiteit Twente

The work in this thesis was carried out at the Physics of Fluids group of the Faculty of Science and Technology of the University of Twente. The research leading to these results has received funding from the [European Community’s] Seventh Framework Programme ([FP7/2007-2013] under grant agreement no215723. This work is also part of the research programme ‘Contact Line Control during Wetting and Dewetting’ (CLC) of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onder-zoek (NWO), ASML and Oc´e.

Publisher:

Tak Shing Chan, Physics of Fluids, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands pof.tnw.utwente.nl

c

⃝ Tak Shing Chan, Enschede, The Netherlands 2012

No part of this work may be reproduced by print photocopy or any other means without the permission in writing from the publisher

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DYNAMICAL WETTING TRANSITIONS: LIQUID FILM

DEPOSITION AND AIR ENTRAINMENT

PROEFSCHRIFT

ter verkrijging van

de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus,

Prof. dr. H. Brinksma,

volgens besluit van het College voor Promoties in het openbaar te verdedigen

op donderdag 30 Augustus 2012 om 16.45 uur

door

Tak Shing Chan geboren op 6thFebruary 1982

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Dit proefschrift is goedgekeurd door de promotor:

Prof. dr. rer. nat. Detlef Lohse

en de assistent-promotor:

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1

Introduction

1.1 Moving contact lines

When you see a water drop moving on a leaf or a window, you might not realize that this very common phenomenon is a challenging fluid mechanical problem, involving physics from molecular to macroscopic scales. The behavior of the moving drop is determined by the dynamics of the contact line, the boundary where two immiscible fluids (liquid/gas or liquid/liquid) and a solid meet. One notices that a moving drop appears round at the front, and thinner and sharper at the rear (see Fig. 1.1). Interest-ingly, detailed experimental studies found that the contact line at the rear develops a corner and this corner becomes sharper as the drop speed increases [1]. Eventually small droplets are released at the tip of the corner if speed is too fast. This instability demonstrates a remarkable property of moving contact lines: the contact line cannot move beyond a maximum speed Ucrelative to the solid surface [2, 3]. If one imposes a speed beyond Uc, there will be a dynamical wetting transition. Like in the moving drop case, the contact line at the rear cannot catch up the speed of the main drop body, thus the liquid at the tip is detached.

Contact line motion is very important in many industrial processes. One exam-ple is in Immersion Lithography [4, 5], where light is directed through a lens on a substrate to construct circuit patterns. A portion of liquid of high refractive index (usually water) is injected between the lens and the substrate and maintained there

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2 CHAPTER 1. INTRODUCTION

Figure 1.1: Drop running down on an inclined plane at different speeds. Drop speed increases from (a) to (h). (a)-(e) The running drop appears round at the front and sharper at the rear. (f)-(h) Droplets are emitted at the tail of the drop. Figures taken from Podgorski et al. [1].

in order to increase the optical resolutions, see Fig. 1.2a. For example, resolution of∼40nm can be achieved when using pure water as the working liquid. Since the substrate cannot be completely covered by the lens, the patterns are not printed on the whole substrate at once. After some patterns have been printed on one region of the substrate, the substrate translates so that an unprinted region is under the cover of the lens. During the translation of the substrate, the contact line is moving relative to the substrate. The problem is that, when the substrate is moving too fast, droplets are emitted at the receding part of the liquid similar to what happens in the moving drop case, see Fig. 1.2b [4]. Similarly, at much higher speed, air bubbles are observed in the advancing part, suggesting entrainment of air at the advancing contact line. All these instabilities could introduce defects to the printed patterns. From an industrial perspective, it is important to push the substrate speed in order to enhance the pro-duction rate and lower the cost while maintaining high quality of their products at the same time. For this it is crucial to understand how to optimize the critical speeds. In other words, what are the relevant physical parameters (e.g. wettability of the

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sur-1.2. HYDRODYNAMIC CHALLENGES 3

;ĂͿ

;ďͿ

^ƉĞĞĚ

Figure 1.2: (a) Sketch of an Immersion Lithography setup. Small droplets and bub-bles appear when the speed is too fast, due to a dynamical wetting transition. (b) Top images are shapes of sliding drops at different speeds as in Fig. 1.1. Bottom images are shapes of water for different speeds of substrate in a setup used in [4] to mimic the Immersion Lithography system. The shapes of the liquids in these two different setups are very similar. Figures taken from Winkels et al.[4].

face, liquid viscosity, surface tension) for the critical speeds of dynamical wetting transitions? How do the critical speeds depend on those parameters? Moreover, the difference in critical speeds of the receding and the advancing parts implies different underlying mechanisms. Why is there a difference? To what extent are the proper-ties of the air relevant for the bubble formation? This thesis therefore addresses the fundamentals of “dynamical wetting transitions”.

1.2 Hydrodynamic challenges

The fluid mechanical description of contact line motion is challenging for various reasons. Firstly, it has been known since Huh and Scriven [6] pointed out in 1971 that, the shear stressτsgoes to infinity when approaching the contact line if a no-slip boundary condition is imposed on the liquid/solid interface. This can be understood by a simple argument. Suppose a liquid drop is moving with typical speed U . If we look at the tail of the drop, thenτs∼ηU/h, where η is the liquid viscosity and h the height of the interface, see Fig. 1.3a. When approaching the contact line, h→ 0, then

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4 CHAPTER 1. INTRODUCTION θǁ h U ;ĂͿ U air liquid ;ďͿ liquid

Figure 1.3: (a) Streamlines in a wedge geometry of one-phase flow when a plate is moving with speed U . (b) Streamlines in a wedge geometry of two-phase flow when a plate is moving from the air to the liquid (exact solutions by Huh & Scriven [6]). Streamlines in the air are very dense when the wedge angleθwis close toπ.

τs→ ∞. This means no drop could move if continuum hydrodynamics with no-slip boundary condition remained valid at the region very close to the contact line. Hence a microscopic mechanism that regularizes the singular tendency at the contact line is required. Many models have been proposed since then that includes precursor film [7], diffusion across the interface [8], Navier-slip condition [6] and others. For an overview we refer the reader to [9]. No matter which model is used, a microscopic length scale has to be introduced to the problem because of the molecular origin of the regularization. This means the typical length scale near the contact line is of order 1nm-10nm.

Above 10nm from the contact line, classical hydrodynamics is valid for the de-scription of the flow in the fluids and the meniscus is determined by the balance between the viscous force and the capillary force [9]. It is natural then to define a dimensionless speed, the capillary number

Ca≡ηU/γ, (1.1)

whereγ is the liquid-gas surface tension. As we move further away from the contact line, reaching macroscopic scales, the viscous effect diminishes. At large scales,

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1.3. DIP COATING GEOMETRY 5

the shape of the interface is determined by the surface tension and the body force, e.g. gravity. In this region, we have macroscopic length scale, e.g. capillary length of order millimeters or the size of the drop. We thus see that there are different length scales involved, characterized by different physical mechanisms. This multi-scale property makes the problem difficult to solve. One difficulty one may think of immediately is that large amount of computation resource in numerical calculations is required to resolve the small scales near the contact line, as also a well-known difficulty in study of turbulence.

A second difficulty of the moving contact line problem is the following. In most of the situations one aims to determine the shape of the fluid-fluid interface and how the interface shape evolves. The interface profiles can be computed by solving normal stress condition at the interface (the Young-Laplace equation) which relates the local curvatureκ of the interface to the normal stress difference across the interface δτn:

γκ = δτn. (1.2)

To evaluateδτn, one needs to determine the velocity fields in the fluids, but these de-pend on the shape of the interface. Solving this coupled problem is very challenging. Thirdly, the standard theoretical tool to deal with problems of flow in thin film is the so-called lubrication theory [10]. When applying it to moving contact line problems, it requires (i) the local slope of the interface to be small, i.e. h′≪ 1 and (ii) the surrounding air viscosity to be negligible. The first condition usually means that the equilibrium contact angleθe has to be small. Experimentally this condition can be fulfilled by properly choosing the material of the substrate and the working fluids. The second condition is more subtle in particular when air entrainment occurs at the advancing contact line. In that case, the local angle of the interface measured in the air is small. One can imagine that the shear stress in the air could be significant since the air is forced to flow in a very confined region. To demonstrate this, we plot the streamlines of flow in a wedge of small angle in air in Fig. 1.3b. We clearly see that the streamlines are extremely dense in the air. This means velocity gradients are large, and thus shear stresses could be significant even though the air viscosity is small. Hence to study air entrainment, one has to consider both the flows in the liquid and the air.

1.3 Dip coating geometry

In this thesis, we wish to reveal the nature of dynamical wetting transitions for re-ceding and advancing contact lines. We focus on the common setup of dip coating, a plate of partially wetting surface (θe> 0) being withdrawn from, or plunged into, a

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6 CHAPTER 1. INTRODUCTION ;ĂͿ ;ďͿ _U θ e z cl liquid air ai U θ e static meniscus z cl liquid air

Figure 1.4: Schematic of the dip coating geometry. (a) When U = 0, the meniscus touches the wall at an static contact angleθe(dashed curve). When the plate is being withdrawn from the reservoir, U̸= 0, the contact line equilibrates above the static equilibrium position (solid curve). Zoom: In this study, we assume the microscopic contact angle at the contact line remains the same as the static contact angleθe. (b) When the plate is plunged, the contact line position zclfalls below its static equilib-rium value.

bath of liquid with speed U as shown in Fig. 1.4. The advantage of this geometry with respect to sliding drops or immersion lithography (Fig. 1.2) is that the flow is two-dimensional so that the description of the problem is largely simplified. This allows for a detailed study of the physical mechanisms underlying the dynamical wettting transitions.

1.3.1 Receding contact lines: film deposition

The dip coating geometry has been studied extensively recently, in particular the receding contact line problem, in which case the plate is withdrawn [3, 11–15]. If the plate is not moving, U = 0, there is no flow in the fluids, and the interface equilibrates to a static shape due to balance between capillarity and gravity. The interface makes an equilibrium angleθe with the solid as a result of intermolecular interaction between the three phases at the contact line. When the plate is moving, the viscous drag generated by the moving plate deforms the fluid interface gradually. After a transient time, the contact line position zclrelaxes to a steady position above its static equilibrium value as long as U is below a critical value Uc, see Fig 1.4a. For U > Uc, a dynamical wetting transition occurs during which a liquid film will be

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1.3. DIP COATING GEOMETRY 7

deposited on the solid surface.

Using lubrication theory [13], one can compute the steady menisci up to the critical capillary number, Cac≡ηUc/γ. To characterize the solutions, the meniscus rise zclis plotted as function of Ca in Fig. 1.5a (solid curve) [14]. We see that zcl increases from the static equilibrium position as Ca increases. After reaching Cac (point (ii) of the solid curve, we refer those solutions before as lower branch), the curve then turns back to Ca < Cacbut with solutions of increasing zcl(these solutions are called upper branch). Further upwards, we observe a series of bifurcations. The shapes of the menisci at different positions of the bifurcation diagram are plotted in Fig. 1.5b. For solutions above point (iii) indicated in the bifurcation diagram, the menisci develop a dimple and above it a ridge-like shape. This ridge grows longer and longer as we move further upward in the bifurcation diagram [12, 16]. Interestingly, in the limit of zcl→ ∞ corresponding to Ca = Ca∗, a film solution referred as “thick film” can be found, see Fig. 1.5c [15]. Again, the thick film solution matches to the bath through a dimple in between. We should emphasize that the thick film solution is different qualitatively from the classical Landau-Levich (LLD) film [17].

The lubrication calculations of the lower branch of the steady solutions have been well verified by experiments for Ca < Ca∗ [14]. For Ca∗< Ca < Cac, the scenario is more complex. Although steady solutions can still be obtained from the lubrication theory, surprisingly, detailed experimental studies found that the dynamical wetting transition starts at Ca∗instead of Cac [12, 14]. Why the critical speed Cacis avoided remains an open question.

Fig. 1.6 shows a photo of an experiment during dynamical wetting transition, we see a ridge is formed right behind the contact line and is propagating upward [12]. We can also observe that the liquid is not entrained at the edge of the plate which has non-zero curvature. From this one may expect the critical speed at the edge of the plate to be higher. Moreover, contrary to the plate case, when withdrawing a fiber of small radius, Sedev and Petrov [18] found transition occurs at Ca = Cac. This raises the question of how the geometry of the flow can influence the dynamical wetting transition.

1.3.2 Advancing contact lines: air entrainment

The reverse process of withdrawing a plate is plunging a plate into a liquid reservoir, see Fig. 1.4b. In this case, we expect air entrainment to occur above a critical velocity. This means the angle in the air gets smaller as Ca increases. Near the dynamical wetting transition (the wedge angle θw close toπ), we argued previously that air viscosity could play a role since air is forced to flow in a very confined region, see Fig. 1.3b. So an interesting question to ask is: is air viscosityηg really relevant for

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8 CHAPTER 1. INTRODUCTION ;cͿ ;ŝͿ ;ŝŝŝͿ ;ŝǀͿ ;ŝŝͿ ;aͿ ZŝĚŐĞƐ ŝŵƉůĞ ;ďͿ

Figure 1.5: (a) Meniscus rise zcl normalized by the capillary length ℓγ versus Ca. Solid curve: steady solutions calculated by lubrication theory, point (ii) corresponds to the maximum, or critical capillary number Cac. Symbols: experimental results obtained during dynamical wetting transition, here the capillary number is defined by the contact line speed with respect to the plate instead of plate speed. For details, we refer to Delon et al. [14]. (b) Stationary interface profiles at different positions of the bifurcation diagram in (a). Note that all the curves have the same bath position. Figures taken from Delon et al. [14]. (c) Red curve: Thick film solution calculated by lubrication equation. Symbols are evolution of the interface observed in experiment during the dynamical wetting transition. Figure taken from Snoeijer et al. [15].

the advancing motion of contact lines?

A very similar scenario for air entrainment without contact line motion [19] has been observed for the case where a viscous liquid impacts a reservoir, as in Fig. 1.7. In this study, a horizontal cylinder partially immersed in a reservoir is rotating with constant speed, thus dragging the liquid on the left out from the bath. This liquid flows along the surface of the cylinder and hits the liquid bath on the right. It has been observed that the air-liquid interface forms a cusp shape at the region of impact (Fig. 1.7a). Above a critical speed, air is entrained (Fig. 1.7b). Lorenceau et al. [19] found that the critical capillary number of air entrainment Cac depends logarithmically on

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1.3. DIP COATING GEOMETRY 9 ŽŶƚĂĐƚůŝŶĞƐ ZŝĚŐĞ dŚĞůŝƋƵŝĚŝƐŶŽƚ ĞŶƚƌĂŝŶĞĚŚĞƌĞ͘ >ŝƋƵŝĚďĂƚŚ WůĂƚĞ ϱϬŵŵ

Figure 1.6: Photograph of the withdrawing plate experiment using a very viscous silicone oil (viscosity = 4.95Pa·s). A ridge is formed right behind the contact line and propagating upward during dynamical wetting transition. Note that while the liquid is entrained at the central region, the contact line has remained still at the edge of the wafer. Figure taken from Snoeijer et al. [12].

the ratio between the air viscosity and the liquid viscosity (R≡ηg/η), Cac≡ηℓ

Uc

γ ∼ −lnR, (1.3)

consistent with the prediction by Eggers’s theory [20]. Does a similar result hold for the plunging plate case in which there is a contact line motion? Using matched asymptotic expansions, Cox extended the study for one phase to two-phase moving contact line problem in a general geometry [21]. He found results similar to the liquid impact case: a weak logarithmic dependence of the critical capillary number on the viscosity ratio R. However, as acknowledged already by Cox, the assumptions of the model break down when Ca is close to Cac. So what do we observe if we investigate the problem experimentally, for example, in the dip-coating geometry?

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10 CHAPTER 1. INTRODUCTION ;ĂͿ ;ďͿ ;ĐͿ ĂŝƌĨŝůŵ

Figure 1.7: (a) A rotating horizontal cylinder is partially immersed in a bath of liquid. The liquid on the left is dragged out from the bath and hits the liquid bath on the right, thus the interface is deformed at the region of impact. (b) Air film (black line) is entrained to the liquid when the cylinder is rotating faster than the critical speed. (c) Critical speed of air entrainment Cacas function of viscosity ratio (on logarithmic scale) between the upper fluid (ηo) and the bottom fluid (η). Figures taken from Lorenceau et al. [19].

More generally, the role of air has recently become a subject of discussion in a wide context of problems. When impacting a drop on a substrate, it was found that splashing can be suppressed completely by reducing the air pressure [22]. Re-markably, in experiments of plunging a tape into a liquid bath, Benkreira et al. [23] observed that the critical speeds of air entrainment can be increased by depressurizing the gas, see Fig. 1.8. Yet, a pressure reduction does not affect the dynamical viscosity of the air. It thus remains unclear how the air plays a role for all these phenomena.

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1.4. GUIDE THROUGH THE THESIS 11

Figure 1.8: Experimental results of plunging a tape into a liquid bath: Critical speed of air entrainment Vae as function of air pressure. Different symbols correspond to different liquid viscosities. Vae is enhanced by depressurizing the air. Figure taken from Benkreira et al. [23].

1.4 Guide through the thesis

In this thesis we investigate dynamical wetting transitions in the dip-coating geometry and address the questions raised above. In Chapter 2 and 3, we investigate liquid film deposition by receding contact lines, while Chapter 4 and 5 address air entrainment by advancing contact lines.

The withdrawing plate problem can be investigated by numerically solving the lubrication equation discussed above. On the other hand, the existence of two sepa-rated length scales (from nanometer to millimeter) allows for analysis using matched asymptotic expansion. This method has been applied to study the withdrawing plate problem by Eggers [3, 11] and provides the lower branch solutions of the bifur-cation diagram and the critical capillary number. In Chapter 2, we will implement this method to study the upper branch solutions and show that the bifurcation at the critical point is a saddle-node type.

To understand how the geometry of the wetting flow influences the dynamical wetting transition, we investigate a problem of withdrawing a fiber from a liquid reservoir in Chapter 3. We calculate the critical speeds and the bifurcation diagrams for different values of fiber radius. We will show that the critical speed decreases with the radius of the fiber. Strikingly, we will demonstrate that the bifurcation diagram for small fiber radius differs qualitatively from that for large fiber radius.

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12 CHAPTER 1. INTRODUCTION

To unravel the role of the air, we experimentally study a plate being plunged into a reservoir of silicon oil of different viscosities in Chapter 4. We will show that the air viscosity plays an important role in air entrainment which is not expected from the classical viewpoint of wetting problems.

In Chapter 5, we will generalize the lubrication model to situations of two-phase flow. This will allow us to investigate air entrainment in the presence of a moving contact line. In particular, we investigate the dynamical wetting transition of air entrainment. The results of the two-phase hydrodynamic theory will be compared with Lattice Boltzmann simulations.

In Chapters 2 - 5, we focus on the regime of small Reynolds number Re ≡ ρUL/η, thus the inertia effect is not taken into consideration. However, some very common fluids (e.g. water) we encounter in environment and industrial applications have a relatively low viscosity. For example, in Immersion Lithography, the fluid used in between the lens and the substrate is water which has viscosity η around 10−3 mPa·s and densityρ = 103kg/m3 at room temperature. If we take the typical speed of flow U as 0.1 m/s and the distance between the lens and the substrate L as 10−4m, then Re = 10. In Chapter 6, we will consider the situation in which Re is not so small. Again we consider a dip-coating geometry. We will develop a lubrication-type model for one-phase flow with the inertial term taken into account.

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2

Theory of the forced wetting transition

∗

We consider a solid plate being withdrawn from a bath of liquid which it does not wet. At low speeds, the meniscus rises below a moving contact line, leaving the rest of the plate dry. At a critical speed of withdrawal, this solution bifurcates into another branch via a saddle-node bifurcation: two branches exist below the critical speed, the lower branch is stable, the upper branch is unstable. The upper branch eventually leads to a solution corresponding to film deposition. We add the local analysis of the upper branch of the bifurcation to a previous analysis of the lower branch. We thus provide a complete description of the dynamical wetting transition in terms of matched asymptotic expansions.

2.1 Introduction

Consider a partially wetting solid plate (with microscopic contact angleθe), being withdrawn from a liquid reservoir, as illustrated in Fig. 2.1. Depending on the speed of withdrawal U , two scenarios can occur. If U is above a certain threshold value

Uc, a liquid film is deposited on the solid surface [2, 9, 12, 15, 24]. This principle is commonly used in the coating industry [25, 26]. On the other hand, if U < Uc , an initially dry solid surface will remain dry, but the contact line position zcl(see

∗_{Published as: T.S. Chan, J.H. Snoeijer, and J. Eggers, Theory of the forced wetting transition, Phys.}

Fluids 24, 072104 (2012).

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14 CHAPTER 2. THEORY OF FORCED WETTING

¯

θ_ap

h(x)

liquid air

U

x

θ

_e

Figure 2.1: Schematic diagram of a plate being withdrawn from a viscous liquid reservoir. The interface shape is h(x), where x is measured relative to the contact line position: x = zcl− z. From a large scale, the interface meets the wall with an apparent contact angleθap. Near the contact line, the interface is highly curved, and one recovers the equilibrium contact angleθe at the contact line.

Fig. 2.1) rises above its equilibrium value. The critical speed is set by a balance between the liquid-gas surface tensionγ and viscous forces [21, 27], which are pro-portional to the liquid viscosityη. As a result, the critical speed is controlled by the dimensionless capillary number Ca≡ηU/γ. For simplicity, we restrict ourselves to the most frequently used geometry of a plate being withdrawn vertically.

The bifurcation between wetted and dry states can be understood by considering the solution curve shown in Fig. 2.2a, which plots zclas function of the plate speed [13, 28]. Here and in the following, all lengths are scaled by the the capillary length

ℓc≡ √

γ/ρg, where ρ is the fluid density and g the acceleration of gravity. Stationary solutions have been computed numerically using a modified lubrication theory [29], which remains valid for arbitrary interface slopes. The only restriction on its validity, when compared to the full viscous fluid equations, is that of small Ca. To model the contact line motion, we introduce a microscopic slip lengthλs, which is necessary for the contact line to be able to move [6, 9]. On the scale ofλs, the interface makes a finite contact angle with the solid, which we take to be the equilibrium contact angle. More details of the modeling will be discussed below.

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2.1. INTRODUCTION 15 0.8 1 1.2 1.4 x 10-4 1.2 1.4 1.6 1.8 2 2.2 2.4 Ca

¯

Ca

c 0 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4 h

¯

0 0.01 0.02 0.03 0.04 0.05 1.2 1.3 1.4 1.5 h ¯ ;ĂͿ ;ďͿ

Figure 2.2: (a) Bifurcation diagram of stationary solutions, showing the meniscus rise zcl(in units of ℓc), as function of Ca forθe= 0.2 and slip lengthλs= 10−5. Solid curve: result from a numerical solution of lubrication theory, see [13]. The horizontal dotted line indicates zcl=

√

2, which is the maximum rise height of a meniscus at equilibrium. (b) Interface profiles of the lower branch solution (solid curve) and the upper branch solution (dashed curve) for Ca = 1.3x10−4(indicated by circles in (a)). The contact line position is at z = zcl, while z = 0 at the bath. In the upper branch, the interface develops a finger close to the wall, making zclmuch larger than expected from the extrapolation of the far-field profile.

Below the horizontal dashed line of Fig. 2.2(a), the contact line position is a monotonically increasing function of speed. For very small Ca, the capillary rise

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approaches its equilibrium value [30]

zcl= √

2(1− sinθe). (2.1)

As the speed increases, the lower branch solution ends at a maximum value Cac of the control parameter, which is the typical scenario for a saddle-node bifurcation [31]. At this point a transition toward a wetted state must occur, since all available steady states correspond to a smaller speed than the actual speed Cac. The lower branch of the solution curve corresponds to a stable equilibrium, the upper branch to an unstable equilibrium [31]. Eventually, the contact line position goes to infinity, and the plate is covered completely by a film.

In [3, 11], the lower branch of the solution curve of Fig. 2.2(a) has been studied analytically, using the method of matched asymptotic expansions. This approach exploits the disparity of scales between the capillary length (≈ 10−3m), and the slip length (10−9m), relevant only very close to the contact line. The outer solution is controlled by a static balance between surface tension and gravity, enabling one to use the relationship (2.1) between capillary rise and contact angle, but with an “apparent” contact angleθap:

zcl= √

2(1− sinθap). (2.2)

The maximum value of zcl consistent with (2.2) is zcl=

√

2, which is realized for θap= 0. Since (2.2) does not allow for solutions above zcl=

√

2, a transition occurs as the apparent contact angle vanishes. The corresponding maximum capillary number Cachas been calculated in [11]. An inspection of Fig. 2.2a shows that the bifurcation indeed occurs very close to zcl =

√

2, and the value of Cac agrees quantitatively with the theoretical prediction, as well as with experiment [13, 14]. An experiment withdrawing a fiber also found the transition to occur at vanishing apparent contact angle [18].

In this Chapter, we will supply the missing description of the upper branch in terms of matched asymptotic expansions. To our knowledge, this is the first time both branches of a saddle-node bifurcation have been described using this method. Matching requires a new type of inner solution, as illustrated in Fig. 2.2(b). We show solutions on the upper and lower branches of the transition (circles in Fig. 2.2(a)), but corresponding to the same value of Ca. It is clear from the main graph of Fig. 2.2(b) that the two solutions are virtually identical on the large scale, and thus correspond to the same value of the apparent contact angleθap. However, the upper branch solution is distinguished by a different inner solution, which features a thin viscous “finger”, only visible in the inset, showing a magnified region very close to the plate. As a result, the actual contact line position zclat the tip of the finger is significantly higher

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2.2. LOWER BRANCH AND CRITICAL SPEED 17

than zclobtained by extrapolating from the outer solution, and which gives zclon the lower branch.

This Chapter is organized as follows: we first recount the technique of matched asymptotic expansion used for finding the lower branch solutions, presented in [3, 11]. We then perform an asymptotic analysis for the upper branch solutions by con-sidering another set of inner solutions, which displays the narrow finger shape as discussed above. We thus show that on the upper branch, the true contact line po-sition is shifted upwards from its apparent value. We demonstrate that the upper and lower branch solutions thus found can be joined at the bifurcation point. In the final discussion, we relate the bifurcation theory approach to the result of matched asymptotic expansions, and discuss remaining unsolved problems.

2.2 Lower branch and critical speed

The lower branch of the bifurcation diagram was calculated in [11] using matched asymptotics, for the case that the angle at which the plate is withdrawn is small. In this section we present a brief summary of this calculation, and adapt it to the case of vertical withdrawal. In the matched asymptotic description, the solution is broken up into an inner and an outer region, and denoted by hin(x) and hout(x), respectively. The variable x is defined relative to the contact line position: z = zcl−x, see Fig. 2.1. The full solution is found by imposing appropriate matching conditions between the two solutions.

2.2.1 Inner solution: lubrication approximation

For simplicity, we restrict ourselves to the case of small equilibrium contact angles,

h′_in(0) =θe≪ 1, so we can use the lubrication approximation close to the contact line [9]. Since the length scale in the inner region is set by the slip lengthλs, which is much smaller than ℓc, gravity is negligible. The corresponding lubrication equation reads [13]:

h′′′_in= 3Ca

h2

in+ 3λshin

. (2.3)

Note that the presence of λs makes for a much weaker singularity as hin vanishes. Forλs= 0, no dynamical solution of (2.3) exists which makes a finite contact angle with the substrate. Thus a finite value ofλs> 0 is needed to allow the contact line to move relative to the substrate. We scale outλsfrom the problem, by introducing the similarity solution hin(x) = 3λsH ( xθe 3λs ) , ξ = xθe 3λs . (2.4)

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With these rescalings (2.3) can be expressed in terms of the dimensionless profile

H(ξ),

H′′′= δ

H2_{+ H}, (2.5)

where we introduced a reduced capillary numberδ = 3Ca/θ_e3. The boundary condi-tions on the plate are

H(0) = 0 and H′(0) = 1. (2.6)

Since (2.5) is a third order differential equation, one more condition is required. This condition will be obtained from the matching to the outer solution.

Away from the contact line, where H≫ 1, (2.5) further reduces to

y′′′= 1

y2, (2.7)

where we have put H(ξ) = δ1/3y(ξ). This equation has an exact solution, whose

properties have been summarized in [32]. In parametric form, a solution with y(0) = 0 reads ξ = 21/3_π_Ai(s) β(αAi(s)+βBi(s)) y =₍_α_Ai(s)+1_β_Bi(s))2 } s∈ [s1,∞[, (2.8)

where Ai and Bi are the two Airy functions [33]. The limitξ → 0 corresponds to

s→ ∞, the opposite limitξ → ∞ to s → s1, where s1is a root of the denominator of (2.8):

αAi(s1) +βBi(s1) = 0. (2.9)

Since the solution extends to s =∞, s1has to be the largest root of (2.9).

The solution y(ξ) is thus characterized by α, β and s1, but only two of these parameters are independent due to (2.9). As detailed in [11], the constantβ can be determined by matching (2.8), which is valid only forξ >_{∼ 1, to a solution of (2.5),} which includes the effect of the cutoff and is thus valid down to the positionξ = 0 of the contact line. It was found that

β2₌_{π exp(−1/(3δ))/2}2/3₊_O(_δ), _(2.10) which eliminates one of the remaining two free parameters. The last parameter s1 will be eliminated below by matching the large scale asymptotics of y(ξ) to the outer solution of the problem. To this end, we need the behavior of y(ξ) for large ξ, which can be obtained from (2.8):

y(ξ) =1

2κyξ

2_{+ b}

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2.2. LOWER BRANCH AND CRITICAL SPEED 19 where κy= ( 21/6β πAi(s1) )2 , by=−2 2/3_Ai′_(s 1) Ai(s1) . (2.12)

2.2.2 Outer solution: liquid reservoir

The outer solution hout(x), valid away from the contact line, is governed by a balance between surface tension and gravity [30]:

κ ≡ h′′out (1 + h′2out)3/2

= zcl− x, (2.13)

whereκ is the curvature of the interface. Remember that x = 0 at the contact line, and x = zcl at the height of the bath. The static balance (2.13) is to be solved subject to the boundary conditions

hout(0) = 0, h′out(0) =θap and h′out(zcl) =∞, (2.14) which contains the apparent contact angleθapas sole parameter. A Taylor expansion of the outer solution leads to

hout(x) = tanθapx + 1 2κapx

2₊_O(_x3)_. _(2.15)

Integrating (2.13) once with respect to x, we obtain

1− h ′ out (1 + h′_out2 )1/2 = 1 2(zcl− x) 2_, _(2.16)

where the boundary condition h′_out → ∞ at the position of the reservoir (x = zcl) is used. Evaluating (2.16) at the contact line position (x = 0) and using the geometrical connection sinθ = h′_out/√1 + h′_out2 as well as (2.13), we find

κap= zcl= √

2(1− sinθap),

as quoted in the introduction. Our main interest in this Chapter is the neighborhood of the bifurcation, i.e. the region of smallθap. Thus for the sake of simplicity we contend ourselves with the leading-order expressions for smallθapand find:

hout(x) =θapx +

1−_√θap/2

2 x

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2.2.3 Matching: lower branch

To match the two solutions on the lower branch we first write the inner solution in term of the original variables,

hin(x) = δ1/3 [ κyθe2x2 6λs + byθex +O (1) ] . (2.18)

Comparing this to (2.17), we find the matching conditions

θap = δ1/3byθe, (2.19) 2−θap = √ 2δ1/3κyθ 2 e 3λs . (2.20)

Adding these two conditions leads to an equation for s1as a function ofδ: 2 θeδ1/3 +2 2/3_Ai′_(s 1) Ai(s1) =2 1/6_exp[_−1/(3_δ)] 3πAi2_(s 1)λs/θe . (2.21)

Once s1is known, one can compute the apparent contact angle θap θe =−2 2/3_δ1/3_Ai′_(s 1) Ai(s1) . (2.22)

Analysis of (2.21) shows [11] that solutions cease to exist above a critical reduced speedδc, for which the apparent contact angle goes to zero. According to (2.22), this occurs when the Airy function takes its global maximum, Ai′(s1) = 0, corresponding to smax=−1.088··· (cf. Fig. 2.3). This gives a critical speed

δc= 1 3 [ ln ( δ1/3 c θe2 25/6₃π(Ai(s max))2λs )]₋₁ ; (2.23)

remember thatδc is related to the critical capillary number by Cac=δcθe3/3. This completes the description of the lower branch of Fig. 2.2a, up to the critical capillary number. However, the analysis so far gives no clue as to what happens beyond the transition, and thus does not explain the nature of the transition.

2.3 Upper branch and bifurcation diagram

We now turn to the upper branch of the bifurcation diagram close to the critical speed. Clearly, this branch cannot be described in terms of (2.2), since zclrises above the

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2.3. UPPER BRANCH AND BIFURCATION DIAGRAM 21 -3 -2 -1 0 1 2 3 -0.4 -0.2 0 0.2 0.4 s 1 A i( s 1 ) S max S₀ lower branch upper branch

Figure 2.3: Airy function Ai(s1). The critical point occurs at smax=−1.088··· where the Airy function takes its maximum value. The lower and upper branches of the bifurcation diagram correspond to s1≥ smaxand s0< s1< smaxrespectively, where

s0=−2.338··· is the rightmost zero of Ai(s1).

maximum value of√2 corresponding toθap= 0. The critical point is attained when

Ai(s1) takes its maximum value, at s1= smax=−1.088··· (cf. Fig. 2.3). The rest of the lower branch corresponds to values s1> smax. This suggests that the upper branch can be described in terms of solutions on the other side of the maximum, s1< smax, and below we will work out this idea.

2.3.1 Matching: upper branch

An inspection of (2.12) shows that byis strictly negative in the domain s0< s1< smax, where s0=−2.338··· is the rightmost zero of the Airy function (cf. Fig. 2.3). Thus we cannot impose the same matching as in the previous section, since (2.19) would imply a negative apparent contact angle. We get around this problem by noting that the large scale asymptotics of the inner solution (2.11) is a parabola that has two zeros, ξ = 0 and −2by/κy. The former coincides with the actual position of the contact line, while we interpret the latter as the ‘apparent’ position of the contact line

ξap=−2by κy

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22 CHAPTER 2. THEORY OF FORCED WETTING 200 400 600 800 1000 1200 0 2000 4000 6000 8000 10000 12000 ξ y ( ξ ) ξ_ap 100 200 300 400 0 200 400 600 800 ξ y( ξ )

Figure 2.4: The profile (2.8) for s1=−2.0 andδ = 0.063 (solid line). The dashed line is the asymptotic form (2.11) of the profile for largeξ; from a large scale the contact line appears to be located atξap ∼ 350. The inset is a zoom on the contact line region, showing a narrow finger which ends at the contact lineξ = 0.

Figure 2.4 illustrates how a typical solution (2.8) (solid line) approaches its asymp-totic limit (2.11) (dashed line), so that the relative distance between the two curves goes to zero for large ξ. Matching the next (constant) term in (2.11) would be achieved only at the next order of the asymptotic expansion. Extrapolating the large-scale solution to the plate position, the contact line appears to be located atξ = ξap, while its actual position is at ξ = 0. This is due to the narrow finger exhibited by solutions y(ξ) for the parameter range s0< s1< smax(cf. inset of Fig. 2.4). This finger was already visible in the numerical solutions shown in Fig. 2.2(b).

Expanding (2.11) relative to the apparent contact line positionξap, we obtain

y (ξ) = 1 2κyξ 2_{+ b} yξ + O(1) = 1 2κy ( ξ − ξap )2 − by ( ξ − ξap ) +O(1). (2.25)

The prefactor of the linear term,−by, is now strictly positive on s0< s1< smaxand can therefore be matched to a ‘shifted’ bath solution:

hout(x) = θap ( x− xap ) +1−√θap/2 2 ( x− xap )2 + O((x− xap)3 ) . (2.26)

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2.3. UPPER BRANCH AND BIFURCATION DIAGRAM 23 δ θap 0.052 0.053 0.054 0.055 0.056 0.057 0.0580 0.05 0.1

Figure 2.5: The apparent contact angleθapas function of δ for lower branch (solid line) and upper branch (dotted line). The two curves meet atδ = δc≈ 0.0578, where θap= 0. The dot-dashed line corresponds to (2.37) below.

Retracing the steps of the previous section, one finds the equation for s1as 2 θeδ1/3− 22/3Ai′(s1) Ai(s1) =2 1/6_exp[_−1/(3_δ)] 3πAi2_(s 1)λs/θe . (2.27)

This differs from (2.21) only by a minus sign in the second term of the left hand side. Similarly, the apparent contact angle follows as

θap θe =2 2/3_δ1/3_Ai′_(s 1) Ai(s1) , (2.28)

with a change in sign. In Fig. 2.5, we plotθapas function ofδ for both lower branch (solid line) and upper branch (dotted line). Both curves meet atδ = δc, where the apparent contact angle vanishes. Thus the apparent contact angle decreases as the transition is approached from below, and then increases again on the upper branch.

2.3.2 Meniscus rise

Let us now express the new solution in terms of the meniscus rise zcland compare to the bifurcation diagram of Fig. 2.2(a). To do so, we need to take into account the

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difference between the real position of the contact line atξ = 0, and the apparent positionξap. This difference comes on top of the meniscus rise of the outer solution, given by (2.2), so that in original variables

zcl= √ 2(1−θap/2) +ξap 3λs θe , (2.29)

once more expanding for smallθap. Rewriting (2.25) in terms of the outer variables

x and h, and comparing it to (2.26), we obtain by=−θap δ1/3 , κy= 3·√2λs δ1/3θ e .

Now using (2.24), one finds

ξap 3λs

θe

=√2θap, (2.30)

giving the contact line position directly in terms ofθap. Thus we finally arrive at:

zcl= √ 2 { 1−θap/2 for zcl≤ √ 2 1 +θap/2 for zcl≥ √ 2. (2.31)

To test (2.31), in Fig. 2.6 we replot zcl as function ofδ for the same parameter values as those of Fig. 2.2(a). The solid curve is the numerical solution of the im-proved lubrication theory, the dashed lines represent (2.31), withθap calculated for the lower and upper branches, respectively. The agreement of the numerical result and the analytical results is very good for both branches.

2.3.3 Comparison to bifurcation theory

Returning to the bifurcation argument presented in the introduction, the transition occurs because the solution curve folds over, and is thus guaranteed to have a local expansion of the form

δ − δc= a1 ( zcl− √ 2 )2 +O ( zcl− √ 2 )3 . (2.32)

To discover the local behavior resulting from the matched asymptotics, we insert the expansion

δ − δc=δ1δs1+δ2δs21, δs1= s1− smax (2.33) into (2.21) and (2.27), respectively. On the basis of (2.32), we would expectδ1 to vanish andδ2to be the same above and below the transition.

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2.3. UPPER BRANCH AND BIFURCATION DIAGRAM 25 0.035 0.04 0.045 0.05 0.055 1.4 1.6 1.8 2 2.2 2.4 2.6

δ

¯

δ*

Figure 2.6: Meniscus rise zclas function ofδ for θe = 0.2 and the slip lengthλs= 10−5. Solid line: Same as in Fig. 2.2a. Dashed lines are (2.31), with θap from (2.21), (2.22) (lower branch) and (2.27),(2.28) (upper branch). For large zcl, the curve approaches a limiting speedδ∗.

Instead, we find

δ1=−

3σ22/3smaxθeδc7/3 2(δc+ 1)

, (2.34)

whereσ = ±1 above and below the transition, respectively; the coefficient δ2 does not have a definite sign. Clearly, the bifurcation curve as predicted by matched asymptotics is not smooth at the bifurcation point. Indeed, close inspection of Fig. 2.6 reveals that the dashed line does not have a vertical tangent at the turning bifurcation point [11]. Instead, the “correct” behavior emerges only in the limit of small slip lengthλs→ 0 orδc→ 0. In this limit, we find

δ2= 3smaxδc2, (2.35)

which is indeed the same above and below the transition. Thus the first term of (2.33) is negligible (and the structure (2.32) is valid) forδs1≫δc1/3.

In addition, from (2.22) and (2.28) we find that to leading order

θap=−22/3θesmaxδc|δs1|, (2.36) which is valid both above (δs1 < 0) and below (δs1 > 0) the transition. Thus in

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26 CHAPTER 2. THEORY OF FORCED WETTING summary we find θap= 22/3θe smax 3 1/2δc−2/3|δ − δc|1/2, (2.37) which is valid for θap≫δc2/3. The asymptotic local behavior (2.37) is plotted in Fig. 2.5 as the dot-dashed line. Its behavior for smallθapis hard to distinguish from (2.22) and (2.28).

2.4 Discussion

Matched asymptotics clearly gives a quantitative description of both branches of the saddle-node bifurcation. In the case of the lower branch, this holds true for the entire branch down to vanishing speed. The behavior of the upper branch, on the other hand, is considerably more complicated, as described in [14, 15]. Following the upper branch, one encounters an infinite sequence of saddle-node bifurcations, as the solution curve oscillates around a second characteristic speed δ∗. The oscillations of the solution curve, which are due to oscillations of the interface profile [16], are damped very quickly, so at a capillary rise of a few times the capillary length the solution effectively corresponds to a film of constant thickness covering the plate. This film, of thickness h∗=√δ∗θ3

e, is maintained by a balance of viscosity and gravity. Experimentally, this film has indeed been realized when a plate is withdrawn with speed aboveδ∗[15].

Our present analysis is not able to capture this feature, since gravity is not in-cluded in the balance (2.3), which describes the inner solution. As the finger seen in the inset of Fig. 2.4 grows in length, the hydrostatic pressure difference across it becomes significant, and the solution fails. An estimate of the capillary rise at which this occurs will be given by the point where the theoretical curve crosses the vertical lineδ∗, where gravity and viscosity balances. Using (2.31) and (2.37), this leads to the estimate zcl− √ 2≈θeδc−2/3 √ δc−δ∗ (2.38)

for the rise at which gravity becomes significant. Usingδc≈ 0.058 andδ∗≈ 0.044, this leads to zcl−

√

2≈ 0.16, which agrees reasonably well with Fig. 2.6.

It is worth reviewing the relative merits of bifurcation theory and those of matched asymptotics, which are complementary. Within bifurcation theory, once one under-stands the origin of the transition as a fold of the solution curve of Fig. 2.2(a), the local structure or order of the transition results automatically. In addition, it is clear that the transition must correspond to moving from a stable branch to an unstable

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2.4. DISCUSSION 27

branch [31]. As always, the disadvantage is that the critical parameters of the transi-tion, such asδc, cannot be calculated within bifurcation theory.

Matched asymptotics, on the other hand, requires a detailed calculation, which reveals the full spatial structure of the solution above and below the transition, as well as the values of all the critical parameters, calculated directly from the system parameters. For example, it is shown that the upper branch solution contains an ad-ditional, and unforeseen structure, which is the finger seen in Fig. 2.2(b). However, the local structure (2.32) of the saddle-node bifurcation emerges from the calculation only in the limitλs→ 0 orδc→ 0. Note the subtle point that the two branches as pre-dicted by asymptotics do not fit together to form a differentiable curve. The reason is that both parts present very different spatial features, and hence agreement between matched asymptotic expansion and bifurcation theory is achieved in a pointwise fash-ion only. It would be an interesting project to see if the next order of the asymptotic expansion will reproduce the next term in the expansion about the bifurcation point.

The stability properties of the two branches have not yet been investigated within matched asymptotics, and remain a non-trivial problem: the task is to properly sep-arate the timescales of the inner and outer solutions. The result is expected to be an effective dynamics [13], in which the solution moves quasistatically along the so-lution curve. This picture of quasi-steady dynamics was confirmed in experiments where the plate velocity was taken above the critical speed [14]. It was found that during the deposition of the liquid film, the upward motion of the contact line fol-lows the bifurcation curve perfectly when replacing the plate velocity by the relative velocity of the contact line, i.e. U− dzcl/dt.

In conclusion, we have calculated the upper and lower branches of a saddle-node bifurcation using matched asymptotics. We are not aware of any other example of this having been done before. It would be of great interest to develop a more general framework of correspondences between certain types of bifurcations, and the matched asymptotics needed to describe them.

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3

Maximum speed of dewetting on a fiber

∗

A solid object can be coated by a nonwetting liquid since a receding contact line can-not exceed a critical speed. We theoretically investigate this forced wetting transition for axisymmetric menisci on fibers of varying radii. First, we use a matched asymp-totic expansion and derive the maximum speed of dewetting. For all radii we find the maximum speed occurs at vanishing apparent contact angle. To further investigate the transition we numerically determine the bifurcation diagram for steady menisci. It is found that the meniscus profiles on thick fibers are smooth, even when there is a film deposited between the bath and the contact line, while profiles on thin fibers exhibit strong oscillations. We discuss how this could lead to different experimental scenarios of film deposition.

3.1 Introduction

A convenient way to deposit a thin liquid layer on a surface is by withdrawing a solid from a liquid reservoir. The film is dragged along with the solid due to the viscous friction of the liquid. This principle is known as dip-coating and is a commonly used technique in industrial contexts [25, 26]. Once deposited on the surface, the film often has a thickness as predicted by Landau, Levich [17] and Derjaguin [34], scaling with

∗_{Published as: T.S. Chan, T. Gueudr´e and J.H. Snoeijer, Maximum speed of dewetting on a fiber,}

Phys. Fluids 23, 112103 (2011).

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30 CHAPTER 3. DEWETTING ON A FIBER

speed U of withdrawal as h∝ U2/3. Recently, however, a different class of solutions were identified, which are much thicker and scale as h ∝ U1/2_{[15]. These thick films} were indeed realized experimentally in the case where the solid was partially wetting. The conditions of partial wetting introduces another interesting feature, namely that the film entrainment only appears above a critical velocity of withdrawal [9, 21, 27, 35, 36]. Below this speed the contact line finds at a steady position, indicated as the meniscus rise∆ (Fig. 3.1). Due to viscous drag between the liquid and the solid, the dynamical position of∆ is higher than at equilibrium. This means that the apparent contact angleθapof the dynamical meniscus is smaller than the equilibrium angleθe. The simplest interpretation of the transition to film deposition is that the apparent contact angle θap→ 0 at a critical plate velocity. This idea was already postulated by Derjaguin and Levi [24], although the energy argument given by de Gennes [35] suggested a nonzero θap at the transition. The hypothesis of θap=0, however, was given a rigorous mathematical basis (for a flat solid) by asymptotic expansions of the lubrication equations [3, 11]. Actually, it was shown by [9] that de Gennes energy argument can be extended to incorporate interface curvature: this exactly gives the lubrication equation, meaning that also the energy argument leads to a zeroθapat the transition. This theory gives a simple prediction for the maximum rise, based on the static meniscus solution with vanishing contact angle – for a fiber of radius r0this simply becomes [30, 37]

∆max≃ { r0(ln4ℓ_r0c− c) for r0≪ ℓc √ 2ℓc for r0≫ ℓc. (3.1)

Here ℓc = (γ/ρg)1/2 is the capillary length based on surface tension γ, density ρ, gravity g and c is Euler’s constant (0.57721). At intermediate radii r0∼ ℓc, the max-imum rise can be determined numerically.

Experimentally, the description of the forced wetting transition has remained ambiguous. The condition of a vanishing apparent contact angle was convincingly shown by Sedev & Petrov [18]. When withdrawing fibers or thin cylinders (r0/ℓc∼ 0.06− 1), they found a maximum rise of the meniscus consistent with (3.1). Using cylinders of larger radii (r0/ℓc ∼ 10), Maleki et al. [38] found zero or nonzeroθap at the transition, depending on the wayθap was determined. When using the crite-rion based on the meniscus height, the transition was found slightly before reaching ∆max. Yet another set of experiments using a flat plate (r0/ℓc=∞) displayed a tran-sition to film depotran-sition clearly before reaching the maximum rise [12, 14]. Still, during the unsteady entrainment phase the maximum recorded speed was reached exactly at√2ℓc. Note that in these experiments, the deposited liquid was not simply

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3.1. INTRODUCTION 31 θ_ap

∆

h

r

o

x

liquid air

U

o

Figure 3.1: Schematic representation of the dip-coating setup: A fiber or cylinder of radius r0is withdrawn with speed U0from a bath of viscous liquid. The axisymmetric meniscus profile is characterized by h(x), while∆ denotes the maximum rise above the reservoir.

the Landau-Levich-Derjaguin film, but gave rise to thick films and even shock solu-tions. It was argued that the presence of these dynamical solutions are related to the pre-critical onset of entrainment [13], but an explanation is still lacking.

An additional complexity is that the contact line can spontaneously develop sharp corner structures, or even zig-zags. This has been observed in dip-coating [2], splash-ing [39], immersion lithography [4, 40] and for drops slidsplash-ing down an inclined plane [1, 41]. The conical structure of the interface near the contact line renders the problem truly three-dimensional, which affects the balance of the capillary forces [42]. For sliding drops, it has been observed experimentally and described by a 3D lubrication model, that this change in geometry indeed leads to a nonzero apparent contact angle at the transition to liquid deposition [43, 44]. This raises the question of how the geometry of the flow can influence the critical speed of wetting [16].

In this Chapter, we theoretically study the withdrawal of fibers of arbitrary radii. By varying the ratio r0/ℓc, we continuously cover the range from thin fibers to the flat plate. First, we extend the asymptotic analysis that was previously done for the flat plate [3, 11] (see also Chapter 2) to the limit of thin fibers (Sec. 3.2). To resolve the singularity of viscous stress near the contact line [6, 45], we introduce a slip length λs[46, 47]. Other types of microscopic regularization will give similar results [9]. Typical values for the slip and capillary lengths are λs∼ 10−9m and ℓc ∼ 10−3m

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respectively. We can thus exploit the hierarchy of length scales

λs≪ r0≪ ℓc, (3.2)

and perform a matched asymptotic expansion. The control parameter is the capillary number Ca = U0η/γ, which is the speed of withdrawal scaled by viscosity η and sur-face tensionγ. The analysis yields the critical capillary number, which depends on the value of r0, and confirms that the maximum speed coincides withθap= 0, for all fiber radii r0. In this sense, the change in geometry does not qualitatively change the nature of the critical point. However, striking differences do show up when comput-ing numerically the complete bifurcation diagrams for all steady solutions (Sec. 3.3). These diagrams include solution branches above ∆maxthat are unstable, but which have been observed as transients during film deposition for the plate case [14]. We find that for small fiber radii much below ℓc, the steady solutions no longer smoothly join the film solutions that mediate the deposition. In the Discussion section we spec-ulate that this is why, experimentally, it is easier to approach the critical point for thin fibers (Sec. 3.4).

3.2 Asymptotic analysis

We compute the shape of an axisymmetric meniscus on a fiber of radius r0using the method of matched asymptotic expansions. The interface is characterized by h(x), as sketched in Fig. 3.1. The matching procedure is outlined schematically in Fig. 3.2. At small scales, the dominant balance is between viscosityη and surface tension γ, and is characterized by the capillary number Ca. Viscous effects can be neglected on large scales, for which the interface profile is that of a static meniscus. The problem is closed by matching the inner and outer solutions. The analysis provides the meniscus rise∆ as a function of Ca as well as the critical speed, both of which can be observed experimentally. We consider both large fiber radii (r0 ≫ ℓc) and small fiber radii (r0≪ ℓc). In all cases we take r0and ℓc to be macroscopic and much greater than the microscopic cutoff. Throughout the analysis, we scale all lengths by the capillary length, i.e. ℓc= 1.

3.2.1 Inner solution: lubrication approximation

To distinguish the solution h in the inner region and the outer region, we denote

hin(x) as the solution in inner region and hout(x) as the solution in outer region. The characteristic length scale for the inner solution comes from the cutoff of the viscous singularity, which here we take the slip lengthλs. Since typical interface curvatures

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3.2. ASYMPTOTIC ANALYSIS 33

κ =0

θ_ap

Inner

region

Matching to outer region

U

o θ_e

Outer region

Gravity +κ=0

Figure 3.2: Schematic diagram showing the different asymptotic regions for the case of a thin fiber. The inner region originates from a balance between viscosity and surface tension. It has a microscopic contact angleθe. The outer region is a static meniscus joining a fiber with an apparent contact angleθap. When the fiber radius r0≪ ℓc, the outer profile is further separated into two regions [37].

turn out∼ Ca1/3/λs, as can be observed from the rescalings below, we can neglect the curvature contribution due to axisymmetry, which is of order 1/r0. Hence, for the inner solution we can follow the analysis by Eggers [3, 11], which was originally derived for the flat plate, and discussed in Chapter 2 as well. For completeness, we briefly summarize the analysis and the central results.

By restricting the analysis to small contact angles, h′_in(0) =θe ≪ 1, one can determine h(x) from the lubrication approximation [10]:

h′′′_in= 3Ca

h2

in+ 3λshin

. (3.3)

Since the slip lengthλsis the only length scale, we rescale the solutions according to hin(x) = 3λsH ( xθe 3λs ) , ξ = xθe 3λs . (3.4) Hence (3.4) reduces to

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H′′′= δ

H2_{+ H}, (3.5)

where we introduced a reduced capillary numberδ = 3Ca/θ_e3. The boundary condi-tions are

H(ξ = 0) = 0, (3.6)

H′(ξ = 0) = 1 (3.7)

and the asymptotic behavior that has to be matched to the outer solution. Away from the contact line, where H≫ 1, (3.5) further reduces to

y′′′= 1

y2, (3.8)

where we have put H(ξ) = δ1/3y(ξ). This equation has an exact solution, whose

properties have been summarized in [32]. In parametric form, a solution with y(0) = 0 reads ξ = 21/3_π_Ai(s) β(αAi(s)+βBi(s)) y =₍_α_Ai(s)+1_β_Bi(s))2 } s∈ [s1,∞[, (3.9)

where Ai and Bi are Airy functions [33]. The limitξ → 0 corresponds to s → ∞, the opposite limitξ → ∞ to s → s1, where s1is a root of the denominator of (3.9):

αAi(s1) +βBi(s1) = 0. (3.10)

Since the solution extends to s =∞, s1has to be the largest root of (3.10).

The solution y(ξ) is thus characterized by three parameters α, β and s1. Note that these are related according to (3.10), so that only two parameters are independent. The constantβ can be determined by matching (3.9), which is valid only for ξ >_{∼ 1,} to a solution of (3.5), which includes the effect of the cutoff and is thus valid down to the positionξ = 0 of the contact line [11]. It was found that

β2₌_{π exp(−1/(3δ))/2}2/3_{+ O(}_δ), _(3.11) which eliminates one of the two free parameters. The remaining parameter will be eliminated below by matching the large scale asymptotics of y(ξ) the outer solution

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3.2. ASYMPTOTIC ANALYSIS 35

of the problem. For that, we only need the asymptotic behavior of y(ξ) for large ξ, which reads: y(ξ) =1 2κyξ 2_{+ b} yξ + O(1), (3.12) where κy= ( 21/6β πAi(s1) )2 , by=−2 2/3_Ai′_(s 1) Ai(s1) . (3.13)

3.2.2 Outer solution: static meniscus

At the scale of outer solution one can neglect viscous effects, and the profile is gov-erned by surface tension and gravity. Thus equating the hydrostatic pressure and the capillary pressure gives

κ = ∆ − x, (3.14)

whereκ is the curvature of the interface. Remind that we expressed all lengths in the capillary length ℓc= 1. The curvature can be expressed from the geometric relation

κ = h ′′ out (1 + h′2 out)3/2 − 1 (r0+ hout)(1 + h′out2 )1/2 . (3.15)

The corresponding outer solution hout(x) is that of a meniscus of a liquid reservoir joining the fiber surface. The contact angle of the meniscus at the surface is denoted as the apparent contact angle,θap, since it refers to the apparent angle on the scale of the outer solution. The boundary conditions therefore are:

hout(x = 0) = 0, (3.16)

h′_out(x = 0) = θap, (3.17)

h′_out(x =∆) = ∞. (3.18)

For the present analysis we require only the asymptotic behavior near the contact line, which is obtained by a Taylor expansion,

hout(x) =θapx + 1 2κapx

2₊_O(_x3)_. _(3.19)

Note that we consider smallθap, since the inner solution is obtained in the lubrication limit.

Dynamical wetting transitions : liquid film deposition and air entrainment

liquid film deposition and

air entrainment

Tak Shing Chan

Dynamical wetting

transitions:

liquid film deposition and

air entrainment

Dynamical wetting transitions:

liquid f

ilm de

position and

air entr

ainment

Dynamical wetting transitions: liquid film deposition and air

entrainment

DYNAMICAL WETTING TRANSITIONS: LIQUID FILM

DEPOSITION AND AIR ENTRAINMENT

Contents

1

Introduction

1.1

Moving contact lines

;ĂͿ

;ďͿ

1.2

Hydrodynamic challenges

1.3

Dip coating geometry

1.4

Guide through the thesis

2

Theory of the forced wetting transition

∗

2.1

Introduction

¯

h(x)

U

x

θ

¯

Ca

¯

2.2

Lower branch and critical speed

2.3

Upper branch and bifurcation diagram

δ

¯

2.4

Discussion

3

Maximum speed of dewetting on a fiber

∗

3.1

Introduction

∆

h

r

x

U

3.2

Asymptotic analysis

κ =0

Inner

region

U

Outer region

Gravity +κ=0